corso di dottorato: ottimizzazione strutturale parte c - franco bontempi
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1Ottimizzazione Strutturale
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Introduzione alla
OTTIMIZZAZIONE STRUTTURALE (parte C)
Franco Bontempi
Ordinario di Tecnica delle Costruzioni
Facolta’ di Ingegneria Civile e Industriale
Sapienza Universita’ di Roma
2
2015
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Object of the course
• Introduction of basic and advanced ideas
and aspects of structural design without to
much stress on the analytical apparatus
but with some insigth on the computational
techniques.
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STRATEGY #0:
DECOMPOSITION
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STRUCTURAL
QUALITY
- design life
- railway
runability
- highway
runability
- free channel
- robustness
- durability
- management
GLOBAL
GEOMETRY
AND
TOPOLOGY
TOPOLOGY
- suspension system
- towers
- towers foundation
- anchor system
- main deck
- deck landing
- ...
GLOBAL GEOMETRY
- main span
- sx span
- dx span
SECTIONAL GEOMETRY
- continuous girder sections
- transverse section
- main cables
- hangers
- towers
- secondary elements
MATERIALS
CHARACTERISTICS
- girders
- cables
SYNTHESIS OF
STRUCTURAL
SOLUTION
AND
DOCUMENTATION
BOUNDARY
CONDITIONS
CONSTRAINTS:
rigid and elastic
constraints,
imposed
displacements
NATURAL
ACTIONS
- temperature
- wind
- earthquake
ANTROPIC
ACTIONS
a) permanent
loading
system
b) variable
- railway
- highway
c) accidental
CO
NV
EN
TIO
NA
L M
OD
ELIN
G:
QU
AS
I ST
AT
IC R
EP
RE
SE
NT
AT
ION
BASIC STRUCTURAL
CONFIGURATION
PARAMETERS
- individuation
- definition
- uncertainty
- description
- bounding
GLOBAL
MODELING
- 2D
- 3D
MODELING WITH
DYNAMIC INTERACTION
ALTERNATIVE STRUCTURAL
CONFIGURATIONS
GLOBAL
OPTIMIZATION
- topology
- morphology
- parametric
LOCAL
OPTIMIZATION
- girders section
- transverse
section
- restraint zone
EXPERT AND
FIXED CHOICES
MEASURES
a) qualitative
b) materials volumes
c) serviceability
- modal characteristics
- deflections
- deformations
- reversibility
d) collapse scenarios
- collapse characteristics
- robustness
e) accidental scenarios
- configurations
- risks
DETAILED
MODELING
EXTENDED
MODELING
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4
5
6
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Numerical Modeling for the
Structural Analysis and Design of
MESSINA STRAIT BRIDGE:
subdivision and development of activities.
FB - june 6, 2005 / [email protected] 7
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STRATEGY #1: SENSITIVITY
governance of priorities
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STRATEGY #2: BOUNDING
behavior governance
p
(p)
p
(p)
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Super
ControlloreProblema Risultato
Solutore #1
Solutore #2
Voting System
STRATEGY #3: REDUNDANCY
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STRUCTURAL
MODELING
CODE
Global Frame Models Local Models
Frame
Work
Substruct-
ured Models
STRUCTURAL
MODELING
CODE
Global Frame Models Local Models
Frame
Work
Substruct-
ured Models
structural configurations
specificity of the modelingcommercial
codes
Design of Experiments (DOE)
• In general usage, design of experiments (DOE) or
experimental design is the design of any information-
gathering exercises where variation is present, whether under
the full control of the experimenter or not. However, in
statistics, these terms are usually used for controlled
experiments.
• Formal planned experimentation is often used in evaluating
physical objects, chemical formulations, structures,
components, and materials. Other types of study, and their
design, are discussed in the articles on computer
experiments, opinion polls and statistical surveys (which are
types of observational study), natural experiments and quasi-
experiments (for example, quasi-experimental design).
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Simulation & Approximation
of the response (≈ surrogate)
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The nature of optimum (2)
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A sub-optimal solution
to a problem is one
that is less than perfect.
Slack situation: loose and not pulled tight.
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OPTIMIZATION METHODS
Heuristics
Nelder – Mead
Genetic Algorithm
Bounded Rationality
Bounded rationality is the idea that in decision-making, rationality
of individuals is limited by the information they have, the
cognitive limitations of their minds, and the finite amount of time
they have to make a decision. It was proposed by Herbert A.
Simon as an alternative basis for the mathematical modeling of
decision making, as used in economics, political science and
related disciplines; it complements rationality as optimization,
which views decision-making as a fully rational process of finding
an optimal choice given the information available. Another way to
look at bounded rationality is that, because decision-makers lack
the ability and resources to arrive at the optimal solution, they
instead apply their rationality only after having greatly simplified
the choices available. Thus the decision-maker is a satisfier, one
seeking a satisfactory solution rather than the optimal one. Ottimizzazione Strutturale
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Model Extensions
• Ariel Rubinstein proposed to model bounded rationality by
explicitly specifying decision-making procedures..
• Gerd Gigerenzer opines that decision theorists have not really
adhered to Simon's original ideas and proposes and shows
that simple heuristics often lead to better decisions than
theoretically optimal procedures.
• Huw Dixon later argues that it may not be necessary to
analyze in detail the process of reasoning underlying bounded
rationality. If we believe that agents will choose an action that
gets them "close" to the optimum, then we can use the notion
of epsilon-optimization, that means you choose your actions
so that the payoff is within epsilon of the optimum. The notion
of strict rationality is then a special case (ε=0).
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εὑρίσκω
• Heuristic (/hjʉˈrɪstɨk/; Greek:
"Εὑρίσκω", "find" or "discover") refers
to experience-based techniques for
problem solving, learning, and
discovery that give a solution which is
not guaranteed to be optimal. Where
the exhaustive search is impractical,
heuristic methods are used to speed
up the process of finding a satisfactory
solution via mental shortcuts to ease
the cognitive load of making a
decision. Examples of this method
include using a rule of thumb, an
educated guess, an intuitive judgment,
stereotyping, or common sense.
• In more precise terms, heuristics are
strategies using readily accessible,
though loosely applicable, information
to control problem solving in human
beings and machines.
• L'euristica (dalla lingua greca εὑρίσκω,
letteralmente "scopro" o "trovo") è una
parte dell'epistemologia e del metodo
scientifico.
• Si definisce procedimento euristico, un
metodo di approccio alla soluzione dei
problemi che non segue un chiaro
percorso, ma che si affida all'intuito e
allo stato temporaneo delle
circostanze, al fine di generare nuova
conoscenza. È opposto al
procedimento algoritmico. In
particolare, l'euristica di una teoria
dovrebbe indicare le strade e le
possibilità da approfondire nel
tentativo di rendere una teoria
progressiva.
Simulated Annealing (Metropolis)
• Simulated annealing (SA) is a generic probabilistic heuristic for the
global optimization problem of locating a good approximation to the
global optimum of a given function in a large search space.
• The name and inspiration come from annealing in metallurgy, a
technique involving heating and controlled cooling of a material to
increase the size of its crystals and reduce their defects.
• This notion of slow cooling is implemented in the Simulated
Annealing algorithm as a slow decrease in the probability of
accepting worse solutions as it explores the solution space.
Accepting worse solutions is a fundamental property of heuristics
because it allows for a more extensive search for the optimum.
• The method is an adaptation of the Metropolis-Hastings algorithm, a
Monte Carlo method to generate sample states of a thermodynamic
system, invented by M.N. Rosenbluth and published in a paper by
N. Metropolis et al. in 1953.
Basic version (2)
Points for SA
• Diameter of the search graph
• Transition probabilities
• Acceptance probabilities
• Efficient candidate generation
• Barrier avoidance
• Cooling schedule
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Nelder-Mead Method (Amoeba)
• The Nelder–Mead method or downhill simplex
method or amoeba method is a commonly used
nonlinear optimization technique, which is a
well-defined numerical method for problems for
which derivatives may not be known.
• The Nelder–Mead technique is a heuristic
search method that was proposed by John
Nelder & Roger Mead (1965) for minimizing an
objective function in a many-dimensional space.
Remarks
;-)
Genetic Algorithm (GA)
• The original motivation for the GA approach was a biological
analogy. In the selective breeding of plants or animals, for example,
offspring are sought that have certain desirable characteristics,
characteristics that are determined at the genetic level by the way
the parents’ chromosomes combine. In the case of GAs, a
population of strings is used, i.e. chromosomes.
• The recombination of strings is carried out using analogies of
genetic crossover and mutation, and the search is guided by the
results of evaluating the objective function f for each string in the
population.
• Based on this evaluation, strings that have higher fitness (i.e.,
represent better solutions) can be identified, and these are given
more opportunity to breed.
Coding
• One of the distinctive features of the GA approach is to
allow the separation of the representation of the problem
from the actual variables in which it was originally
formulated. In line with biological usage of the terms, it
has become customary to distinguish the ‘genotype’—
the encoded representation of the variables, from the
‘phenotype’—the set of variables themselves.
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Genotype space = {0,1}L
(mappa)
Phenotype space
(territorio)
Encoding
(representation)
Decoding
(inverse representation)
01101001
01001001
10010010
10010001
Translation
Tensile crack phenomena in HCS
(splitting, bursting, spalling).
• splitting cracks: caused by stresses resulting from
the development of prestressing in the anchorage
zone, that may generate traction stresses in the
concrete.
• bursting cracks: a local effect, generated by the
strand slippage into the slab end while the former
widens slightly on being cut.
• spalling cracks: occurring above the axis of the
strands in the HCS end zone, caused also by the
development of prestressing in the concrete at the
slab ends where only the lower part holding the
strands begins to be prestressed.Ottimizzazione Strutturale
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Tensile crack phenomena in HCS
(splitting, bursting, spalling).
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Cross-section of the reference HCS
and numerical model
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Tensile deformations in the vertical
directions for the spalling effect
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The binary coding of the geometry
characteristics of the section
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• The fitness function F includes terms to represent the weight of the slab.
• First, functions gi(x), represents the geometry constraints, implicitly satisfied during the definition of the variable space.
• Functions hi(x) represent the structural safety constraints. In this study, two checks are carried out:1. the first one on the bending stress, carried out after the
initial structural analyses on the meso-scale model. 2. the second one, on the spalling stress, carried out on the
micro-scale model.
• If both checks are positive, the individual is fitting the constrain conditions, otherwise, it is discarded and a different element is introduced in the population.
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Original values values obtained
after the optimization process
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1° Step
2° Step
Limit States
Service Limit States Ultimate Limit States
Prestressed Continuous Beam
Elements of nonlinear formulation
Equilibrium Equations
Nominal behavior
Level of uncertainty
Uncertainty
α - level
Random / Optimized Sampling
Cujaba River Bridge
Cujaba River Bridge
Ultimate Limit States (ULS)
Uses of genetic algorithm
• To perform the stochastic exploration of the load space;
• To handle the uncertainties related to the definition of the loads;
• To investigate the global behavior of the structure by means of the definition of the envelope diagram of the performances;
• To define the worst load combination;
• To scrutinize the exact value of a specific performance.
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Dissipation devices
Soil behaviorMaterial
Soil-Structure interface Contact
Support system
Pylon
Cable system
Geometrical
Soil-Structure
Response
Vehicle-StructureWind-Structure
Nonlinearities
Interactions
Action
Uncertainties
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Dependability attributes threats,
means and their interactions.
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Performance in relation to the
return periods of the actions.
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S N
Geometry of the long-span
suspension bridge considered.
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A genetic algorithm approach for
performance assessment• The performance of a long-span suspension bridge is
investigated by means of a GA approach. • Focus is given to three aspects of the structural behavior of
the bridge:1) maximum vertical displacement;2) maximum longitudinal and transversal slope;3) maximum tension in main cable and in the tower legs.
• The load scenarios that lead to the most severe performance metrics are explored in the space of the load variables by an optimization process based on GA’s.
• The implementation of a GA based optimization is essential since the traditional optimization techniques are rather ineffective, due to the high number of dimensions of the load variables space and the presence of numerous local optimum points.
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Loading systems considered in the
genetic algorithm analysis.
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Remarks on loading system
• Traffic and train loads are directed vertically but the possibility to have a longitudinal component due to the acceleration (A) or the deceleration (D) is also taken into account.
• In addition, a torque is present if the traffic loads are not positioned on the axis of the respective box girder section.
• The wind action, assumed always present and flowing transversally to the longitudinal axis of the bridge, produces lift, drag and torque.
• In order to represent analytically the entire loading system, 16 variables are needed.
• Since each of the girders is formed by 123 finite elements, the position of the loads will be defined by integer variables, ranging, in general, from 0 to 123.
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Variables considered for the
definition of the loading system.
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Binary coding
• The position variables are implemented in binary
code with a dimension of eight bits (the minimum
dimension able to represent the position of the
loads on the bridge deck):
• In this row vector, x1 defines the position of the
train on the bridge deck, in binary code: for
example, if the train load starts from the fifteenth
element on the deck, the variable x1 is:
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Population
• The GA starts by considering an initial population of N
row vector x created assigning random values to the
unknown variables; each row of the matrix X represents
the chromosome of one solution:
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Target functions
• In order to evaluate the performance of the bridge, the
following six target functions are considered:
1. the vertical displacement (negative) for the bridge deck;
2. the horizontal displacement (positive) for the bridge
deck;
3. the longitudinal slope for the bridge deck;
4. the transversal slope for the bridge deck;
5. the axial tension for the main cables;
6. the stress state induced by the axial action and the two
bending moments for the bridge tower legs.
• Each performance is measured by the peak value over
all nodes of the bridge deck. Ottimizzazione Strutturale
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Evolution of population
• For each target function, the genetic algorithm creates new populations of N row vector x in order to find the worse configurations of the considered loads.
• The genetic algorithm works by evaluating the target function in correspondence with each assumed vector x.
• If the population contains a N number of x vectors, the best N/2 vectors are saved in a new population while the other vectors are erased.
• To complete the new population, additional N/2 vectors are formed from the saved vectors using the genetic operator of mutation and crossover.
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Mutation
• The mutation on the generic vector i of the population n changes a single bit of a randomly selected chromosome; for example provides the change from 1 to 0:
• As a result a new vector k is obtained for the population n+1.
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Crossover
• The crossover on the generic vectors i and j of
population n is provided in this example:
• where a group of cells of chromosomes i and j is
selected and the respective states changed.
• As a result there are two new vectors k and l for the
population n+1. Ottimizzazione Strutturale
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Remarks
• When N/2 new vectors are created, the genetic algorithm restarts with the evaluation of the target function for each vector xn+1.
• It should be observed that a genetic algorithm is a stochastic evolutionary procedure because the operators of mutation and crossover are not deterministic but there is a probability of occurrence for each operator.
• Usually the probability of occurrence of the mutation operator is low (0 – 5%) while the probability of occurrence of the crossover operator is high (70 – 90%).
• What makes this procedure attractive is the fact that usually there is a large interdependence between the quality of results and of the choice of these parameters.
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• The FE model consists of 1614 elements (beams, no compression cable elements and gaps) and 1140 nodes.
• For each of the six previously chosen performance metrics (target functions), GA analysis is performed with an initial randomly chosen population of 100 chromosomes. For each chromosome the structural analysis, accounting for geometrical and material nonlinearities, is developed using ADINA, starting each time from the reference configurations under permanent loads and adding the traffic and wind loads.
• The custom software reads the output evaluation and performs the genetic recombination of the chromosomes to get a new generation of chromosomes: 100 cycles of regeneration are considered for a total of 10000 different load scenarios, each of them leading to a nonlinear structural analysis.
• The probability of occurrence of the crossover operator is of 80% while the probability of occurrence of the mutation operator is of 2%.
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Remarks
• It is clear that the convergence of the variables that
define the train position (A) is better than the one that
defines the position of the light traffic load (B).
• From a design point of view, it means that the influence
of the railway load in defining the vertical displacement is
much higher than the traffic load.
• In addition, it can be observed that the railway loads
converge towards two different edges (North and South).
This is due to the fact that the geometry of the bridge is
almost symmetrical.
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INDEX
• Knowledge
• Limits
• Scale effects
• Ergonomy
• PeopleOttimizzazione Strutturale
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CONOSCENZA
RICHIESTA
DA UN PROGETTO
EVOLUTIVO
CONOSCENZA
RICHESTA
DA UN PROGETTO
INNOVATIVO
BASE DI
CONOSCENZA
ATTUALE
La crescita di conoscenze
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Knowledge Growth Process
KNOWLEDGE
REQUIRED
BY AN EVOLUTIVE
DESIGN
NEW KNOWLEDGE
REQUIRED BY
AN INNOVATIVE
DESIGN
ACTUAL
KNOWLEDGE BASIS
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ST
RU
TT
UR
E C
ON
CO
MP
OR
TA
ME
NT
O P
ER
FO
RM
A
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ST
RU
TT
UR
E C
ON
CO
MP
OR
TA
ME
NT
O V
ET
TO
RIA
LE
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ST
RU
TT
UR
E C
ON
CO
MP
OR
TA
ME
NT
O S
EZ
ION
ALE
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ST
RU
TT
UR
E C
ON
CO
MP
OR
TA
ME
NT
O D
I S
UP
ER
FIC
IE
Dalian, June 2008 173
Causes of system failure
100%
Time
% o
f fa
ilure
Unknown phenomena
Known phenomena
Research
level
Design code
level
past present future
A
BB B
C
Hu
man
err
ors
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Dalian, June 2008 192
Dalian, June 2008 193
0
500
1000
1500
2000
2500
3000
3500
SPAN
SPAN 1100 1298 1385 1410 1624 1991 3300
BISA
N-
VER
RAZZ
JIAN
GYN
HUM
BER
GRE
AT
AKA
SHI
MES
SINA
Dalian, June 2008 194
Dalian, June 2008 195
Dalian, June 2008 210
GLOBAL LEVEL
3300 mLocal level
200 m
Example: Size Effect
Nonaka & Takeuchi:
conoscenza esplicite e implicite
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